In a known face seal assembly according to GB-A 2 286 020 of a type having a pair of annular sealing elements and a eccentric annular space or clearance between a shaft and the non-rotating annular sealing element the eccentric space in which a buffer liquid can be introduced has an axial dimension which differs only insignificantly from the cross-sectional area of inlet and outlet ports through which the buffer liquid is introduced into or discharged from the annular space, respectively. Accordingly the cooling and lubricating effect of the buffer liquid on the annular sealing elements is relatively low. In the U.S. Pat. No. 3,746,349 although the eccentric annular space is elongated, tests carried out within the scope of the present invention have shown that there is still need of a substantial improvement in the cooling and lubricating capacity, since due to prevailing centripetal forces gases contained in and released from the buffer liquid when pressurized in the eccentric annular space may collect in the narrow portions of the annular space, resulting in a corresponding constriction of the cross-sectional flow area so that a flow of the buffer liquid along the annular space will be affected or even totally prevented. From E. Meyer "Axiale Gleitring-dichtungen", published by VDI-Verlag, Dusseldorf/Germany, 1966, it is further known to provide in a stationary member an eccentric pressure chamber for pumping a buffer liquid. The chamber is part of a pumping device which functions as a centrifugal pump in that the eccentric chamber cooperates with a pumping ring concentrically disposed on the rotating sealing element and having a number of vane-type pumping elements. The pumping ring can affect the performance of rotating sealing element and, furthermore, unwanted turbulences of the buffer liquid may occur in portions of the mechanical face seal assembly which necessitate good cooling and lubrication.